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multidimensional.jl
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multidimensional.jl
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# This file is a part of Julia. License is MIT: https://julialang.org/license
### Multidimensional iterators
module IteratorsMD
import .Base: eltype, length, size, first, last, in, getindex,
setindex!, IndexStyle, min, max, zero, oneunit, isless, eachindex,
ndims, IteratorSize, convert, show, iterate, promote_rule, to_indices
import .Base: +, -, *, (:)
import .Base: simd_outer_range, simd_inner_length, simd_index, setindex
using .Base: IndexLinear, IndexCartesian, AbstractCartesianIndex, fill_to_length, tail,
ReshapedArray, ReshapedArrayLF, OneTo
using .Base.Iterators: Reverse, PartitionIterator
using .Base: @propagate_inbounds
export CartesianIndex, CartesianIndices
"""
CartesianIndex(i, j, k...) -> I
CartesianIndex((i, j, k...)) -> I
Create a multidimensional index `I`, which can be used for
indexing a multidimensional array `A`. In particular, `A[I]` is
equivalent to `A[i,j,k...]`. One can freely mix integer and
`CartesianIndex` indices; for example, `A[Ipre, i, Ipost]` (where
`Ipre` and `Ipost` are `CartesianIndex` indices and `i` is an
`Int`) can be a useful expression when writing algorithms that
work along a single dimension of an array of arbitrary
dimensionality.
A `CartesianIndex` is sometimes produced by [`eachindex`](@ref), and
always when iterating with an explicit [`CartesianIndices`](@ref).
# Examples
```jldoctest
julia> A = reshape(Vector(1:16), (2, 2, 2, 2))
2×2×2×2 Array{Int64, 4}:
[:, :, 1, 1] =
1 3
2 4
[:, :, 2, 1] =
5 7
6 8
[:, :, 1, 2] =
9 11
10 12
[:, :, 2, 2] =
13 15
14 16
julia> A[CartesianIndex((1, 1, 1, 1))]
1
julia> A[CartesianIndex((1, 1, 1, 2))]
9
julia> A[CartesianIndex((1, 1, 2, 1))]
5
```
"""
struct CartesianIndex{N} <: AbstractCartesianIndex{N}
I::NTuple{N,Int}
CartesianIndex{N}(index::NTuple{N,Integer}) where {N} = new(index)
end
CartesianIndex(index::NTuple{N,Integer}) where {N} = CartesianIndex{N}(index)
CartesianIndex(index::Integer...) = CartesianIndex(index)
CartesianIndex{N}(index::Vararg{Integer,N}) where {N} = CartesianIndex{N}(index)
# Allow passing tuples smaller than N
CartesianIndex{N}(index::Tuple) where {N} = CartesianIndex{N}(fill_to_length(index, 1, Val(N)))
CartesianIndex{N}(index::Integer...) where {N} = CartesianIndex{N}(index)
CartesianIndex{N}() where {N} = CartesianIndex{N}(())
# Un-nest passed CartesianIndexes
CartesianIndex(index::Union{Integer, CartesianIndex}...) = CartesianIndex(flatten(index))
flatten(I::Tuple{}) = I
flatten(I::Tuple{Any}) = I
flatten(I::Tuple{<:CartesianIndex}) = I[1].I
@inline flatten(I) = _flatten(I...)
@inline _flatten() = ()
@inline _flatten(i, I...) = (i, _flatten(I...)...)
@inline _flatten(i::CartesianIndex, I...) = (i.I..., _flatten(I...)...)
CartesianIndex(index::Tuple{Vararg{Union{Integer, CartesianIndex}}}) = CartesianIndex(index...)
show(io::IO, i::CartesianIndex) = (print(io, "CartesianIndex"); show(io, i.I))
# length
length(::CartesianIndex{N}) where {N} = N
length(::Type{CartesianIndex{N}}) where {N} = N
# indexing
getindex(index::CartesianIndex, i::Integer) = index.I[i]
Base.get(A::AbstractArray, I::CartesianIndex, default) = get(A, I.I, default)
eltype(::Type{T}) where {T<:CartesianIndex} = eltype(fieldtype(T, :I))
# access to index tuple
Tuple(index::CartesianIndex) = index.I
Base.setindex(x::CartesianIndex,i,j) = CartesianIndex(Base.setindex(Tuple(x),i,j))
# equality
Base.:(==)(a::CartesianIndex{N}, b::CartesianIndex{N}) where N = a.I == b.I
# zeros and ones
zero(::CartesianIndex{N}) where {N} = zero(CartesianIndex{N})
zero(::Type{CartesianIndex{N}}) where {N} = CartesianIndex(ntuple(Returns(0), Val(N)))
oneunit(::CartesianIndex{N}) where {N} = oneunit(CartesianIndex{N})
oneunit(::Type{CartesianIndex{N}}) where {N} = CartesianIndex(ntuple(Returns(1), Val(N)))
# arithmetic, min/max
@inline (-)(index::CartesianIndex{N}) where {N} =
CartesianIndex{N}(map(-, index.I))
@inline (+)(index1::CartesianIndex{N}, index2::CartesianIndex{N}) where {N} =
CartesianIndex{N}(map(+, index1.I, index2.I))
@inline (-)(index1::CartesianIndex{N}, index2::CartesianIndex{N}) where {N} =
CartesianIndex{N}(map(-, index1.I, index2.I))
@inline min(index1::CartesianIndex{N}, index2::CartesianIndex{N}) where {N} =
CartesianIndex{N}(map(min, index1.I, index2.I))
@inline max(index1::CartesianIndex{N}, index2::CartesianIndex{N}) where {N} =
CartesianIndex{N}(map(max, index1.I, index2.I))
@inline (*)(a::Integer, index::CartesianIndex{N}) where {N} = CartesianIndex{N}(map(x->a*x, index.I))
@inline (*)(index::CartesianIndex, a::Integer) = *(a,index)
# comparison
@inline isless(I1::CartesianIndex{N}, I2::CartesianIndex{N}) where {N} = _isless(0, I1.I, I2.I)
@inline function _isless(ret, I1::NTuple{N,Int}, I2::NTuple{N,Int}) where N
newret = ifelse(ret==0, icmp(I1[N], I2[N]), ret)
_isless(newret, Base.front(I1), Base.front(I2))
end
_isless(ret, ::Tuple{}, ::Tuple{}) = ifelse(ret==1, true, false)
icmp(a, b) = ifelse(isless(a,b), 1, ifelse(a==b, 0, -1))
# conversions
convert(::Type{T}, index::CartesianIndex{1}) where {T<:Number} = convert(T, index[1])
convert(::Type{T}, index::CartesianIndex) where {T<:Tuple} = convert(T, index.I)
# hashing
const cartindexhash_seed = UInt == UInt64 ? 0xd60ca92f8284b8b0 : 0xf2ea7c2e
function Base.hash(ci::CartesianIndex, h::UInt)
h += cartindexhash_seed
for i in ci.I
h = hash(i, h)
end
return h
end
# nextind and prevind with CartesianIndex
function Base.nextind(a::AbstractArray{<:Any,N}, i::CartesianIndex{N}) where {N}
iter = CartesianIndices(axes(a))
# might overflow
I = inc(i.I, iter.indices)
return I
end
function Base.prevind(a::AbstractArray{<:Any,N}, i::CartesianIndex{N}) where {N}
iter = CartesianIndices(axes(a))
# might underflow
I = dec(i.I, iter.indices)
return I
end
Base._ind2sub(t::Tuple, ind::CartesianIndex) = Tuple(ind)
# Iteration over the elements of CartesianIndex cannot be supported until its length can be inferred,
# see #23719
Base.iterate(::CartesianIndex) =
error("iteration is deliberately unsupported for CartesianIndex. Use `I` rather than `I...`, or use `Tuple(I)...`")
# Iteration
"""
CartesianIndices(sz::Dims) -> R
CartesianIndices((istart:[istep:]istop, jstart:[jstep:]jstop, ...)) -> R
Define a region `R` spanning a multidimensional rectangular range
of integer indices. These are most commonly encountered in the
context of iteration, where `for I in R ... end` will return
[`CartesianIndex`](@ref) indices `I` equivalent to the nested loops
for j = jstart:jstep:jstop
for i = istart:istep:istop
...
end
end
Consequently these can be useful for writing algorithms that
work in arbitrary dimensions.
CartesianIndices(A::AbstractArray) -> R
As a convenience, constructing a `CartesianIndices` from an array makes a
range of its indices.
!!! compat "Julia 1.6"
The step range method `CartesianIndices((istart:istep:istop, jstart:[jstep:]jstop, ...))`
requires at least Julia 1.6.
# Examples
```jldoctest
julia> foreach(println, CartesianIndices((2, 2, 2)))
CartesianIndex(1, 1, 1)
CartesianIndex(2, 1, 1)
CartesianIndex(1, 2, 1)
CartesianIndex(2, 2, 1)
CartesianIndex(1, 1, 2)
CartesianIndex(2, 1, 2)
CartesianIndex(1, 2, 2)
CartesianIndex(2, 2, 2)
julia> CartesianIndices(fill(1, (2,3)))
2×3 CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}:
CartesianIndex(1, 1) CartesianIndex(1, 2) CartesianIndex(1, 3)
CartesianIndex(2, 1) CartesianIndex(2, 2) CartesianIndex(2, 3)
```
## Conversion between linear and cartesian indices
Linear index to cartesian index conversion exploits the fact that a
`CartesianIndices` is an `AbstractArray` and can be indexed linearly:
```jldoctest
julia> cartesian = CartesianIndices((1:3, 1:2))
3×2 CartesianIndices{2, Tuple{UnitRange{Int64}, UnitRange{Int64}}}:
CartesianIndex(1, 1) CartesianIndex(1, 2)
CartesianIndex(2, 1) CartesianIndex(2, 2)
CartesianIndex(3, 1) CartesianIndex(3, 2)
julia> cartesian[4]
CartesianIndex(1, 2)
julia> cartesian = CartesianIndices((1:2:5, 1:2))
3×2 CartesianIndices{2, Tuple{StepRange{Int64, Int64}, UnitRange{Int64}}}:
CartesianIndex(1, 1) CartesianIndex(1, 2)
CartesianIndex(3, 1) CartesianIndex(3, 2)
CartesianIndex(5, 1) CartesianIndex(5, 2)
julia> cartesian[2, 2]
CartesianIndex(3, 2)
```
## Broadcasting
`CartesianIndices` support broadcasting arithmetic (+ and -) with a `CartesianIndex`.
!!! compat "Julia 1.1"
Broadcasting of CartesianIndices requires at least Julia 1.1.
```jldoctest
julia> CIs = CartesianIndices((2:3, 5:6))
2×2 CartesianIndices{2, Tuple{UnitRange{Int64}, UnitRange{Int64}}}:
CartesianIndex(2, 5) CartesianIndex(2, 6)
CartesianIndex(3, 5) CartesianIndex(3, 6)
julia> CI = CartesianIndex(3, 4)
CartesianIndex(3, 4)
julia> CIs .+ CI
2×2 CartesianIndices{2, Tuple{UnitRange{Int64}, UnitRange{Int64}}}:
CartesianIndex(5, 9) CartesianIndex(5, 10)
CartesianIndex(6, 9) CartesianIndex(6, 10)
```
For cartesian to linear index conversion, see [`LinearIndices`](@ref).
"""
struct CartesianIndices{N,R<:NTuple{N,OrdinalRange{Int, Int}}} <: AbstractArray{CartesianIndex{N},N}
indices::R
end
CartesianIndices(::Tuple{}) = CartesianIndices{0,typeof(())}(())
function CartesianIndices(inds::NTuple{N,OrdinalRange{<:Integer, <:Integer}}) where {N}
indices = map(r->convert(OrdinalRange{Int, Int}, r), inds)
CartesianIndices{N, typeof(indices)}(indices)
end
CartesianIndices(index::CartesianIndex) = CartesianIndices(index.I)
CartesianIndices(inds::NTuple{N,Union{<:Integer,OrdinalRange{<:Integer}}}) where {N} =
CartesianIndices(map(_convert2ind, inds))
CartesianIndices(A::AbstractArray) = CartesianIndices(axes(A))
_convert2ind(sz::Bool) = Base.OneTo(Int8(sz))
_convert2ind(sz::Integer) = Base.OneTo(sz)
_convert2ind(sz::AbstractUnitRange) = first(sz):last(sz)
_convert2ind(sz::OrdinalRange) = first(sz):step(sz):last(sz)
"""
(:)(start::CartesianIndex, [step::CartesianIndex], stop::CartesianIndex)
Construct [`CartesianIndices`](@ref) from two `CartesianIndex` and an optional step.
!!! compat "Julia 1.1"
This method requires at least Julia 1.1.
!!! compat "Julia 1.6"
The step range method start:step:stop requires at least Julia 1.6.
# Examples
```jldoctest
julia> I = CartesianIndex(2,1);
julia> J = CartesianIndex(3,3);
julia> I:J
2×3 CartesianIndices{2, Tuple{UnitRange{Int64}, UnitRange{Int64}}}:
CartesianIndex(2, 1) CartesianIndex(2, 2) CartesianIndex(2, 3)
CartesianIndex(3, 1) CartesianIndex(3, 2) CartesianIndex(3, 3)
julia> I:CartesianIndex(1, 2):J
2×2 CartesianIndices{2, Tuple{StepRange{Int64, Int64}, StepRange{Int64, Int64}}}:
CartesianIndex(2, 1) CartesianIndex(2, 3)
CartesianIndex(3, 1) CartesianIndex(3, 3)
```
"""
(:)(I::CartesianIndex{N}, J::CartesianIndex{N}) where N =
CartesianIndices(map((i,j) -> i:j, Tuple(I), Tuple(J)))
(:)(I::CartesianIndex{N}, S::CartesianIndex{N}, J::CartesianIndex{N}) where N =
CartesianIndices(map((i,s,j) -> i:s:j, Tuple(I), Tuple(S), Tuple(J)))
promote_rule(::Type{CartesianIndices{N,R1}}, ::Type{CartesianIndices{N,R2}}) where {N,R1,R2} =
CartesianIndices{N,Base.indices_promote_type(R1,R2)}
convert(::Type{Tuple{}}, R::CartesianIndices{0}) = ()
for RT in (OrdinalRange{Int, Int}, StepRange{Int, Int}, AbstractUnitRange{Int})
@eval convert(::Type{NTuple{N,$RT}}, R::CartesianIndices{N}) where {N} =
map(x->convert($RT, x), R.indices)
end
convert(::Type{NTuple{N,AbstractUnitRange}}, R::CartesianIndices{N}) where {N} =
convert(NTuple{N,AbstractUnitRange{Int}}, R)
convert(::Type{NTuple{N,UnitRange{Int}}}, R::CartesianIndices{N}) where {N} =
UnitRange{Int}.(convert(NTuple{N,AbstractUnitRange}, R))
convert(::Type{NTuple{N,UnitRange}}, R::CartesianIndices{N}) where {N} =
UnitRange.(convert(NTuple{N,AbstractUnitRange}, R))
convert(::Type{Tuple{Vararg{AbstractUnitRange{Int}}}}, R::CartesianIndices{N}) where {N} =
convert(NTuple{N,AbstractUnitRange{Int}}, R)
convert(::Type{Tuple{Vararg{AbstractUnitRange}}}, R::CartesianIndices) =
convert(Tuple{Vararg{AbstractUnitRange{Int}}}, R)
convert(::Type{Tuple{Vararg{UnitRange{Int}}}}, R::CartesianIndices{N}) where {N} =
convert(NTuple{N,UnitRange{Int}}, R)
convert(::Type{Tuple{Vararg{UnitRange}}}, R::CartesianIndices) =
convert(Tuple{Vararg{UnitRange{Int}}}, R)
convert(::Type{CartesianIndices{N,R}}, inds::CartesianIndices{N}) where {N,R} =
CartesianIndices(convert(R, inds.indices))
# equality
Base.:(==)(a::CartesianIndices{N}, b::CartesianIndices{N}) where N =
all(map(==, a.indices, b.indices))
Base.:(==)(a::CartesianIndices, b::CartesianIndices) = false
# AbstractArray implementation
Base.axes(iter::CartesianIndices{N,R}) where {N,R} = map(Base.axes1, iter.indices)
Base.IndexStyle(::Type{CartesianIndices{N,R}}) where {N,R} = IndexCartesian()
@propagate_inbounds function Base.getindex(iter::CartesianIndices{N,R}, I::Vararg{Int, N}) where {N,R}
CartesianIndex(getindex.(iter.indices, I))
end
ndims(R::CartesianIndices) = ndims(typeof(R))
ndims(::Type{CartesianIndices{N}}) where {N} = N
ndims(::Type{CartesianIndices{N,TT}}) where {N,TT} = N
eachindex(::IndexCartesian, A::AbstractArray) = CartesianIndices(axes(A))
@inline function eachindex(::IndexCartesian, A::AbstractArray, B::AbstractArray...)
axsA = axes(A)
Base._all_match_first(axes, axsA, B...) || Base.throw_eachindex_mismatch_indices(IndexCartesian(), axes(A), axes.(B)...)
CartesianIndices(axsA)
end
eltype(::Type{CartesianIndices{N}}) where {N} = CartesianIndex{N}
eltype(::Type{CartesianIndices{N,TT}}) where {N,TT} = CartesianIndex{N}
IteratorSize(::Type{<:CartesianIndices{N}}) where {N} = Base.HasShape{N}()
@inline function iterate(iter::CartesianIndices)
iterfirst = first(iter)
if !all(map(in, iterfirst.I, iter.indices))
return nothing
end
iterfirst, iterfirst
end
@inline function iterate(iter::CartesianIndices, state)
valid, I = __inc(state.I, iter.indices)
valid || return nothing
return CartesianIndex(I...), CartesianIndex(I...)
end
# increment & carry
@inline function inc(state, indices)
_, I = __inc(state, indices)
return CartesianIndex(I...)
end
# Unlike ordinary ranges, CartesianIndices continues the iteration in the next column when the
# current column is consumed. The implementation is written recursively to achieve this.
# `iterate` returns `Union{Nothing, Tuple}`, we explicitly pass a `valid` flag to eliminate
# the type instability inside the core `__inc` logic, and this gives better runtime performance.
__inc(::Tuple{}, ::Tuple{}) = false, ()
@inline function __inc(state::Tuple{Int}, indices::Tuple{<:OrdinalRange})
rng = indices[1]
I = state[1] + step(rng)
valid = __is_valid_range(I, rng) && state[1] != last(rng)
return valid, (I, )
end
@inline function __inc(state, indices)
rng = indices[1]
I = state[1] + step(rng)
if __is_valid_range(I, rng) && state[1] != last(rng)
return true, (I, tail(state)...)
end
valid, I = __inc(tail(state), tail(indices))
return valid, (first(rng), I...)
end
@inline __is_valid_range(I, rng::AbstractUnitRange) = I in rng
@inline function __is_valid_range(I, rng::OrdinalRange)
if step(rng) > 0
lo, hi = first(rng), last(rng)
else
lo, hi = last(rng), first(rng)
end
lo <= I <= hi
end
# 0-d cartesian ranges are special-cased to iterate once and only once
iterate(iter::CartesianIndices{0}, done=false) = done ? nothing : (CartesianIndex(), true)
size(iter::CartesianIndices) = map(length, iter.indices)
length(iter::CartesianIndices) = prod(size(iter))
# make CartesianIndices a multidimensional range
Base.step(iter::CartesianIndices) = CartesianIndex(map(step, iter.indices))
first(iter::CartesianIndices) = CartesianIndex(map(first, iter.indices))
last(iter::CartesianIndices) = CartesianIndex(map(last, iter.indices))
# When used as indices themselves, CartesianIndices can simply become its tuple of ranges
@inline to_indices(A, inds, I::Tuple{CartesianIndices, Vararg{Any}}) =
to_indices(A, inds, (I[1].indices..., tail(I)...))
# but preserve CartesianIndices{0} as they consume a dimension.
@inline to_indices(A, inds, I::Tuple{CartesianIndices{0},Vararg{Any}}) =
(first(I), to_indices(A, inds, tail(I))...)
@inline in(i::CartesianIndex, r::CartesianIndices) = false
@inline in(i::CartesianIndex{N}, r::CartesianIndices{N}) where {N} = all(map(in, i.I, r.indices))
simd_outer_range(iter::CartesianIndices{0}) = iter
function simd_outer_range(iter::CartesianIndices)
CartesianIndices(tail(iter.indices))
end
simd_inner_length(iter::CartesianIndices{0}, ::CartesianIndex) = 1
simd_inner_length(iter::CartesianIndices, I::CartesianIndex) = Base.length(iter.indices[1])
simd_index(iter::CartesianIndices{0}, ::CartesianIndex, I1::Int) = first(iter)
@propagate_inbounds function simd_index(iter::CartesianIndices, Ilast::CartesianIndex, I1::Int)
CartesianIndex(getindex(iter.indices[1], I1+first(Base.axes1(iter.indices[1]))), Ilast.I...)
end
# Split out the first N elements of a tuple
@inline function split(t, V::Val)
ref = ntuple(Returns(true), V) # create a reference tuple of length N
_split1(t, ref), _splitrest(t, ref)
end
@inline _split1(t, ref) = (t[1], _split1(tail(t), tail(ref))...)
@inline _splitrest(t, ref) = _splitrest(tail(t), tail(ref))
# exit either when we've exhausted the input or reference tuple
_split1(::Tuple{}, ::Tuple{}) = ()
_split1(::Tuple{}, ref) = ()
_split1(t, ::Tuple{}) = ()
_splitrest(::Tuple{}, ::Tuple{}) = ()
_splitrest(t, ::Tuple{}) = t
_splitrest(::Tuple{}, ref) = ()
@inline function split(I::CartesianIndex, V::Val)
i, j = split(I.I, V)
CartesianIndex(i), CartesianIndex(j)
end
function split(R::CartesianIndices, V::Val)
i, j = split(R.indices, V)
CartesianIndices(i), CartesianIndices(j)
end
# reversed CartesianIndices iteration
Base.reverse(iter::CartesianIndices) = CartesianIndices(reverse.(iter.indices))
@inline function iterate(r::Reverse{<:CartesianIndices})
iterfirst = last(r.itr)
if !all(map(in, iterfirst.I, r.itr.indices))
return nothing
end
iterfirst, iterfirst
end
@inline function iterate(r::Reverse{<:CartesianIndices}, state)
valid, I = __dec(state.I, r.itr.indices)
valid || return nothing
return CartesianIndex(I...), CartesianIndex(I...)
end
# decrement & carry
@inline function dec(state, indices)
_, I = __dec(state, indices)
return CartesianIndex(I...)
end
# decrement post check to avoid integer overflow
@inline __dec(::Tuple{}, ::Tuple{}) = false, ()
@inline function __dec(state::Tuple{Int}, indices::Tuple{<:OrdinalRange})
rng = indices[1]
I = state[1] - step(rng)
valid = __is_valid_range(I, rng) && state[1] != first(rng)
return valid, (I,)
end
@inline function __dec(state, indices)
rng = indices[1]
I = state[1] - step(rng)
if __is_valid_range(I, rng) && state[1] != first(rng)
return true, (I, tail(state)...)
end
valid, I = __dec(tail(state), tail(indices))
return valid, (last(rng), I...)
end
# 0-d cartesian ranges are special-cased to iterate once and only once
iterate(iter::Reverse{<:CartesianIndices{0}}, state=false) = state ? nothing : (CartesianIndex(), true)
function Base.LinearIndices(inds::CartesianIndices{N,R}) where {N,R<:NTuple{N, AbstractUnitRange}}
LinearIndices{N,R}(inds.indices)
end
function Base.LinearIndices(inds::CartesianIndices)
indices = inds.indices
if all(x->step(x)==1, indices)
indices = map(rng->first(rng):last(rng), indices)
LinearIndices{length(indices), typeof(indices)}(indices)
else
# Given the fact that StepRange 1:2:4 === 1:2:3, we lost the original size information
# and thus cannot calculate the correct linear indices when the steps are not 1.
throw(ArgumentError("LinearIndices for $(typeof(inds)) with non-1 step size is not yet supported."))
end
end
# This is currently needed because converting to LinearIndices is only available when steps are
# all 1
# NOTE: this is only a temporary patch and could be possibly removed when StepRange support to
# LinearIndices is done
function Base.collect(inds::CartesianIndices{N, R}) where {N,R<:NTuple{N, AbstractUnitRange}}
Base._collect_indices(axes(inds), inds)
end
function Base.collect(inds::CartesianIndices)
dest = Array{eltype(inds), ndims(inds)}(undef, size(inds))
i = 0
@inbounds for a in inds
dest[i+=1] = a
end
dest
end
# array operations
Base.intersect(a::CartesianIndices{N}, b::CartesianIndices{N}) where N =
CartesianIndices(intersect.(a.indices, b.indices))
# Views of reshaped CartesianIndices are used for partitions — ensure these are fast
const CartesianPartition{T<:CartesianIndex, P<:CartesianIndices, R<:ReshapedArray{T,1,P}} = SubArray{T,1,R,Tuple{UnitRange{Int}},false}
eltype(::Type{PartitionIterator{T}}) where {T<:ReshapedArrayLF} = SubArray{eltype(T), 1, T, Tuple{UnitRange{Int}}, true}
eltype(::Type{PartitionIterator{T}}) where {T<:ReshapedArray} = SubArray{eltype(T), 1, T, Tuple{UnitRange{Int}}, false}
Iterators.IteratorEltype(::Type{<:PartitionIterator{T}}) where {T<:ReshapedArray} = Iterators.IteratorEltype(T)
eltype(::Type{PartitionIterator{T}}) where {T<:OneTo} = UnitRange{eltype(T)}
eltype(::Type{PartitionIterator{T}}) where {T<:Union{UnitRange, StepRange, StepRangeLen, LinRange}} = T
Iterators.IteratorEltype(::Type{<:PartitionIterator{T}}) where {T<:Union{OneTo, UnitRange, StepRange, StepRangeLen, LinRange}} = Iterators.IteratorEltype(T)
@inline function iterate(iter::CartesianPartition)
isempty(iter) && return nothing
f = first(iter)
return (f, (f, 1))
end
@inline function iterate(iter::CartesianPartition, (state, n))
n >= length(iter) && return nothing
I = IteratorsMD.inc(state.I, iter.parent.parent.indices)
return I, (I, n+1)
end
@inline function simd_outer_range(iter::CartesianPartition)
# In general, the Cartesian Partition might start and stop in the middle of the outer
# dimensions — thus the outer range of a CartesianPartition is itself a
# CartesianPartition.
t = tail(iter.parent.parent.indices)
ci = CartesianIndices(t)
li = LinearIndices(t)
return @inbounds view(ci, li[tail(iter[1].I)...]:li[tail(iter[end].I)...])
end
function simd_outer_range(iter::CartesianPartition{CartesianIndex{2}})
# But for two-dimensional Partitions the above is just a simple one-dimensional range
# over the second dimension; we don't need to worry about non-rectangular staggers in
# higher dimensions.
return @inbounds CartesianIndices((iter[1][2]:iter[end][2],))
end
@inline function simd_inner_length(iter::CartesianPartition, I::CartesianIndex)
inner = iter.parent.parent.indices[1]
@inbounds fi = iter[1].I
@inbounds li = iter[end].I
inner_start = I.I == tail(fi) ? fi[1] : first(inner)
inner_end = I.I == tail(li) ? li[1] : last(inner)
return inner_end - inner_start + 1
end
@inline function simd_index(iter::CartesianPartition, Ilast::CartesianIndex, I1::Int)
# I1 is the 0-based distance from the first dimension's offest
offset = first(iter.parent.parent.indices[1]) # (this is 1 for 1-based arrays)
# In the first column we need to also add in the iter's starting point (branchlessly)
f = @inbounds iter[1]
startoffset = (Ilast.I == tail(f.I))*(f[1] - 1)
CartesianIndex((I1 + offset + startoffset, Ilast.I...))
end
end # IteratorsMD
using .IteratorsMD
## Bounds-checking with CartesianIndex
# Disallow linear indexing with CartesianIndex
function checkbounds(::Type{Bool}, A::AbstractArray, i::Union{CartesianIndex, AbstractArray{<:CartesianIndex}})
@_inline_meta
checkbounds_indices(Bool, axes(A), (i,))
end
@inline checkbounds_indices(::Type{Bool}, ::Tuple{}, I::Tuple{CartesianIndex,Vararg{Any}}) =
checkbounds_indices(Bool, (), (I[1].I..., tail(I)...))
@inline checkbounds_indices(::Type{Bool}, IA::Tuple{Any}, I::Tuple{CartesianIndex,Vararg{Any}}) =
checkbounds_indices(Bool, IA, (I[1].I..., tail(I)...))
@inline checkbounds_indices(::Type{Bool}, IA::Tuple, I::Tuple{CartesianIndex,Vararg{Any}}) =
checkbounds_indices(Bool, IA, (I[1].I..., tail(I)...))
# Indexing into Array with mixtures of Integers and CartesianIndices is
# extremely performance-sensitive. While the abstract fallbacks support this,
# codegen has extra support for SIMDification that sub2ind doesn't (yet) support
@propagate_inbounds getindex(A::Array, i1::Union{Integer, CartesianIndex}, I::Union{Integer, CartesianIndex}...) =
A[to_indices(A, (i1, I...))...]
@propagate_inbounds setindex!(A::Array, v, i1::Union{Integer, CartesianIndex}, I::Union{Integer, CartesianIndex}...) =
(A[to_indices(A, (i1, I...))...] = v; A)
# Support indexing with an array of CartesianIndex{N}s
# Here we try to consume N of the indices (if there are that many available)
# The first two simply handle ambiguities
@inline function checkbounds_indices(::Type{Bool}, ::Tuple{},
I::Tuple{AbstractArray{CartesianIndex{N}},Vararg{Any}}) where N
checkindex(Bool, (), I[1]) & checkbounds_indices(Bool, (), tail(I))
end
@inline function checkbounds_indices(::Type{Bool}, IA::Tuple{Any},
I::Tuple{AbstractArray{CartesianIndex{0}},Vararg{Any}})
checkbounds_indices(Bool, IA, tail(I))
end
@inline function checkbounds_indices(::Type{Bool}, IA::Tuple{Any},
I::Tuple{AbstractArray{CartesianIndex{N}},Vararg{Any}}) where N
checkindex(Bool, IA, I[1]) & checkbounds_indices(Bool, (), tail(I))
end
@inline function checkbounds_indices(::Type{Bool}, IA::Tuple,
I::Tuple{AbstractArray{CartesianIndex{N}},Vararg{Any}}) where N
IA1, IArest = IteratorsMD.split(IA, Val(N))
checkindex(Bool, IA1, I[1]) & checkbounds_indices(Bool, IArest, tail(I))
end
@inline function checkbounds_indices(::Type{Bool}, IA::Tuple{},
I::Tuple{AbstractArray{Bool,N},Vararg{Any}}) where N
return checkbounds_indices(Bool, IA, (LogicalIndex(I[1]), tail(I)...))
end
@inline function checkbounds_indices(::Type{Bool}, IA::Tuple,
I::Tuple{AbstractArray{Bool,N},Vararg{Any}}) where N
return checkbounds_indices(Bool, IA, (LogicalIndex(I[1]), tail(I)...))
end
function checkindex(::Type{Bool}, inds::Tuple, I::AbstractArray{<:CartesianIndex})
b = true
for i in I
b &= checkbounds_indices(Bool, inds, (i,))
end
b
end
checkindex(::Type{Bool}, inds::Tuple, I::CartesianIndices) = all(checkindex.(Bool, inds, I.indices))
# combined count of all indices, including CartesianIndex and
# AbstractArray{CartesianIndex}
# rather than returning N, it returns an NTuple{N,Bool} so the result is inferrable
@inline index_ndims(i1, I...) = (true, index_ndims(I...)...)
@inline function index_ndims(i1::CartesianIndex, I...)
(map(Returns(true), i1.I)..., index_ndims(I...)...)
end
@inline function index_ndims(i1::AbstractArray{CartesianIndex{N}}, I...) where N
(ntuple(Returns(true), Val(N))..., index_ndims(I...)...)
end
index_ndims() = ()
# combined dimensionality of all indices
# rather than returning N, it returns an NTuple{N,Bool} so the result is inferrable
@inline index_dimsum(i1, I...) = (index_dimsum(I...)...,)
@inline index_dimsum(::Colon, I...) = (true, index_dimsum(I...)...)
@inline index_dimsum(::AbstractArray{Bool}, I...) = (true, index_dimsum(I...)...)
@inline function index_dimsum(::AbstractArray{<:Any,N}, I...) where N
(ntuple(Returns(true), Val(N))..., index_dimsum(I...)...)
end
index_dimsum() = ()
# Recursively compute the lengths of a list of indices, without dropping scalars
index_lengths() = ()
@inline index_lengths(::Real, rest...) = (1, index_lengths(rest...)...)
@inline index_lengths(A::AbstractArray, rest...) = (length(A), index_lengths(rest...)...)
# shape of array to create for getindex() with indices I, dropping scalars
# returns a Tuple{Vararg{AbstractUnitRange}} of indices
index_shape() = ()
@inline index_shape(::Real, rest...) = index_shape(rest...)
@inline index_shape(A::AbstractArray, rest...) = (axes(A)..., index_shape(rest...)...)
"""
LogicalIndex(mask)
The `LogicalIndex` type is a special vector that simply contains all indices I
where `mask[I]` is true. This specialized type does not support indexing
directly as doing so would require O(n) lookup time. `AbstractArray{Bool}` are
wrapped with `LogicalIndex` upon calling [`to_indices`](@ref).
"""
struct LogicalIndex{T, A<:AbstractArray{Bool}} <: AbstractVector{T}
mask::A
sum::Int
LogicalIndex{T,A}(mask::A) where {T,A<:AbstractArray{Bool}} = new(mask, count(mask))
end
LogicalIndex(mask::AbstractVector{Bool}) = LogicalIndex{Int, typeof(mask)}(mask)
LogicalIndex(mask::AbstractArray{Bool, N}) where {N} = LogicalIndex{CartesianIndex{N}, typeof(mask)}(mask)
LogicalIndex{Int}(mask::AbstractArray) = LogicalIndex{Int, typeof(mask)}(mask)
size(L::LogicalIndex) = (L.sum,)
length(L::LogicalIndex) = L.sum
collect(L::LogicalIndex) = [i for i in L]
show(io::IO, r::LogicalIndex) = print(io,collect(r))
print_array(io::IO, X::LogicalIndex) = print_array(io, collect(X))
# Iteration over LogicalIndex is very performance-critical, but it also must
# support arbitrary AbstractArray{Bool}s with both Int and CartesianIndex.
# Thus the iteration state contains an index iterator and its state. We also
# keep track of the count of elements since we already know how many there
# should be -- this way we don't need to look at future indices to check done.
@inline function iterate(L::LogicalIndex{Int})
r = LinearIndices(L.mask)
iterate(L, (1, r))
end
@inline function iterate(L::LogicalIndex{<:CartesianIndex})
r = CartesianIndices(axes(L.mask))
iterate(L, (1, r))
end
@propagate_inbounds function iterate(L::LogicalIndex, s)
# We're looking for the n-th true element, using iterator r at state i
n = s[1]
n > length(L) && return nothing
#unroll once to help inference, cf issue #29418
idx, i = iterate(tail(s)...)
s = (n+1, s[2], i)
L.mask[idx] && return (idx, s)
while true
idx, i = iterate(tail(s)...)
s = (n+1, s[2], i)
L.mask[idx] && return (idx, s)
end
end
# When wrapping a BitArray, lean heavily upon its internals.
@inline function iterate(L::Base.LogicalIndex{Int,<:BitArray})
L.sum == 0 && return nothing
Bc = L.mask.chunks
return iterate(L, (1, @inbounds Bc[1]))
end
@inline function iterate(L::Base.LogicalIndex{Int,<:BitArray}, s)
Bc = L.mask.chunks
i1, c = s
while c==0
i1 % UInt >= length(Bc) % UInt && return nothing
i1 += 1
@inbounds c = Bc[i1]
end
tz = trailing_zeros(c) + 1
c = _blsr(c)
return ((i1-1)<<6 + tz, (i1, c))
end
@inline checkbounds(::Type{Bool}, A::AbstractArray, I::LogicalIndex{<:Any,<:AbstractArray{Bool,1}}) =
eachindex(IndexLinear(), A) == eachindex(IndexLinear(), I.mask)
@inline checkbounds(::Type{Bool}, A::AbstractArray, I::LogicalIndex) = axes(A) == axes(I.mask)
@inline checkindex(::Type{Bool}, indx::AbstractUnitRange, I::LogicalIndex) = (indx,) == axes(I.mask)
checkindex(::Type{Bool}, inds::Tuple, I::LogicalIndex) = checkbounds_indices(Bool, inds, axes(I.mask))
ensure_indexable(I::Tuple{}) = ()
@inline ensure_indexable(I::Tuple{Any, Vararg{Any}}) = (I[1], ensure_indexable(tail(I))...)
@inline ensure_indexable(I::Tuple{LogicalIndex, Vararg{Any}}) = (collect(I[1]), ensure_indexable(tail(I))...)
# In simple cases, we know that we don't need to use axes(A). Optimize those
# until Julia gets smart enough to elide the call on its own:
@inline to_indices(A, I::Tuple{Vararg{Union{Integer, CartesianIndex}}}) = to_indices(A, (), I)
# But some index types require more context spanning multiple indices
# CartesianIndexes are simple; they just splat out
@inline to_indices(A, inds, I::Tuple{CartesianIndex, Vararg{Any}}) =
to_indices(A, inds, (I[1].I..., tail(I)...))
# But for arrays of CartesianIndex, we just skip the appropriate number of inds
@inline function to_indices(A, inds, I::Tuple{AbstractArray{CartesianIndex{N}}, Vararg{Any}}) where N
_, indstail = IteratorsMD.split(inds, Val(N))
(to_index(A, I[1]), to_indices(A, indstail, tail(I))...)
end
# And boolean arrays behave similarly; they also skip their number of dimensions
@inline function to_indices(A, inds, I::Tuple{AbstractArray{Bool, N}, Vararg{Any}}) where N
_, indstail = IteratorsMD.split(inds, Val(N))
(to_index(A, I[1]), to_indices(A, indstail, tail(I))...)
end
# As an optimization, we allow trailing Array{Bool} and BitArray to be linear over trailing dimensions
@inline to_indices(A, inds, I::Tuple{Union{Array{Bool,N}, BitArray{N}}}) where {N} =
(_maybe_linear_logical_index(IndexStyle(A), A, I[1]),)
_maybe_linear_logical_index(::IndexStyle, A, i) = to_index(A, i)
_maybe_linear_logical_index(::IndexLinear, A, i) = LogicalIndex{Int}(i)
# Colons get converted to slices by `uncolon`
@inline to_indices(A, inds, I::Tuple{Colon, Vararg{Any}}) =
(uncolon(inds, I), to_indices(A, _maybetail(inds), tail(I))...)
const CI0 = Union{CartesianIndex{0}, AbstractArray{CartesianIndex{0}}}
uncolon(inds::Tuple{}, I::Tuple{Colon, Vararg{Any}}) = Slice(OneTo(1))
uncolon(inds::Tuple, I::Tuple{Colon, Vararg{Any}}) = Slice(inds[1])
### From abstractarray.jl: Internal multidimensional indexing definitions ###
getindex(x::Number, i::CartesianIndex{0}) = x
getindex(t::Tuple, i::CartesianIndex{1}) = getindex(t, i.I[1])
# These are not defined on directly on getindex to avoid
# ambiguities for AbstractArray subtypes. See the note in abstractarray.jl
@inline function _getindex(l::IndexStyle, A::AbstractArray, I::Union{Real, AbstractArray}...)
@boundscheck checkbounds(A, I...)
return _unsafe_getindex(l, _maybe_reshape(l, A, I...), I...)
end
# But we can speed up IndexCartesian arrays by reshaping them to the appropriate dimensionality:
_maybe_reshape(::IndexLinear, A::AbstractArray, I...) = A
_maybe_reshape(::IndexCartesian, A::AbstractVector, I...) = A
@inline _maybe_reshape(::IndexCartesian, A::AbstractArray, I...) = __maybe_reshape(A, index_ndims(I...))
@inline __maybe_reshape(A::AbstractArray{T,N}, ::NTuple{N,Any}) where {T,N} = A
@inline __maybe_reshape(A::AbstractArray, ::NTuple{N,Any}) where {N} = reshape(A, Val(N))
function _unsafe_getindex(::IndexStyle, A::AbstractArray, I::Vararg{Union{Real, AbstractArray}, N}) where N
# This is specifically not inlined to prevent excessive allocations in type unstable code
shape = index_shape(I...)
dest = similar(A, shape)
map(unsafe_length, axes(dest)) == map(unsafe_length, shape) || throw_checksize_error(dest, shape)
_unsafe_getindex!(dest, A, I...) # usually a generated function, don't allow it to impact inference result
return dest
end
function _generate_unsafe_getindex!_body(N::Int)
quote
@_inline_meta
D = eachindex(dest)
Dy = iterate(D)
@inbounds @nloops $N j d->I[d] begin
# This condition is never hit, but at the moment
# the optimizer is not clever enough to split the union without it
Dy === nothing && return dest
(idx, state) = Dy
dest[idx] = @ncall $N getindex src j
Dy = iterate(D, state)
end
return dest
end
end
# Always index with the exactly indices provided.
@generated function _unsafe_getindex!(dest::AbstractArray, src::AbstractArray, I::Vararg{Union{Real, AbstractArray}, N}) where N
_generate_unsafe_getindex!_body(N)
end
# manually written-out specializations for 1 and 2 arguments to save compile time
@eval function _unsafe_getindex!(dest::AbstractArray, src::AbstractArray, I::Vararg{Union{Real, AbstractArray},1})
$(_generate_unsafe_getindex!_body(1))
end
@eval function _unsafe_getindex!(dest::AbstractArray, src::AbstractArray, I::Vararg{Union{Real, AbstractArray},2})
$(_generate_unsafe_getindex!_body(2))
end
@noinline throw_checksize_error(A, sz) = throw(DimensionMismatch("output array is the wrong size; expected $sz, got $(size(A))"))
## setindex! ##
function _setindex!(l::IndexStyle, A::AbstractArray, x, I::Union{Real, AbstractArray}...)
@_inline_meta
@boundscheck checkbounds(A, I...)
_unsafe_setindex!(l, _maybe_reshape(l, A, I...), x, I...)
A
end
function _generate_unsafe_setindex!_body(N::Int)
quote
x′ = unalias(A, x)
@nexprs $N d->(I_d = unalias(A, I[d]))
idxlens = @ncall $N index_lengths I
@ncall $N setindex_shape_check x′ (d->idxlens[d])
Xy = iterate(x′)
@inbounds @nloops $N i d->I_d begin
# This is never reached, but serves as an assumption for
# the optimizer that it does not need to emit error paths
Xy === nothing && break
(val, state) = Xy
@ncall $N setindex! A val i
Xy = iterate(x′, state)
end
A
end
end
@generated function _unsafe_setindex!(::IndexStyle, A::AbstractArray, x, I::Vararg{Union{Real,AbstractArray}, N}) where N
_generate_unsafe_setindex!_body(N)
end
@eval function _unsafe_setindex!(::IndexStyle, A::AbstractArray, x, I::Vararg{Union{Real,AbstractArray},1})
$(_generate_unsafe_setindex!_body(1))
end
@eval function _unsafe_setindex!(::IndexStyle, A::AbstractArray, x, I::Vararg{Union{Real,AbstractArray},2})
$(_generate_unsafe_setindex!_body(2))
end
diff(a::AbstractVector) = diff(a, dims=1)
"""
diff(A::AbstractVector)
diff(A::AbstractArray; dims::Integer)
Finite difference operator on a vector or a multidimensional array `A`. In the
latter case the dimension to operate on needs to be specified with the `dims`
keyword argument.
!!! compat "Julia 1.1"
`diff` for arrays with dimension higher than 2 requires at least Julia 1.1.
# Examples
```jldoctest
julia> a = [2 4; 6 16]
2×2 Matrix{Int64}:
2 4
6 16
julia> diff(a, dims=2)
2×1 Matrix{Int64}:
2
10
julia> diff(vec(a))
3-element Vector{Int64}:
4
-2
12
```
"""
function diff(a::AbstractArray{T,N}; dims::Integer) where {T,N}
require_one_based_indexing(a)
1 <= dims <= N || throw(ArgumentError("dimension $dims out of range (1:$N)"))
r = axes(a)
r0 = ntuple(i -> i == dims ? UnitRange(1, last(r[i]) - 1) : UnitRange(r[i]), N)
r1 = ntuple(i -> i == dims ? UnitRange(2, last(r[i])) : UnitRange(r[i]), N)
return view(a, r1...) .- view(a, r0...)
end
function diff(r::AbstractRange{T}; dims::Integer=1) where {T}
dims == 1 || throw(ArgumentError("dimension $dims out of range (1:1)"))
return T[@inbounds r[i+1] - r[i] for i in firstindex(r):lastindex(r)-1]
end
### from abstractarray.jl
# In the common case where we have two views into the same parent, aliasing checks
# are _much_ easier and more important to get right
function mightalias(A::SubArray{T,<:Any,P}, B::SubArray{T,<:Any,P}) where {T,P}
if !_parentsmatch(A.parent, B.parent)
# We cannot do any better than the usual dataids check
return !_isdisjoint(dataids(A), dataids(B))
end
# Now we know that A.parent === B.parent. This means that the indices of A
# and B are the same length and indexing into the same dimensions. We can
# just walk through them and check for overlaps: O(ndims(A)). We must finally
# ensure that the indices don't alias with either parent
return _indicesmightoverlap(A.indices, B.indices) ||
!_isdisjoint(dataids(A.parent), _splatmap(dataids, B.indices)) ||
!_isdisjoint(dataids(B.parent), _splatmap(dataids, A.indices))
end
_parentsmatch(A::AbstractArray, B::AbstractArray) = A === B
# Two reshape(::Array)s of the same size aren't `===` because they have different headers
_parentsmatch(A::Array, B::Array) = pointer(A) == pointer(B) && size(A) == size(B)
_indicesmightoverlap(A::Tuple{}, B::Tuple{}) = true
_indicesmightoverlap(A::Tuple{}, B::Tuple) = error("malformed subarray")
_indicesmightoverlap(A::Tuple, B::Tuple{}) = error("malformed subarray")
# For ranges, it's relatively cheap to construct the intersection
@inline function _indicesmightoverlap(A::Tuple{AbstractRange, Vararg{Any}}, B::Tuple{AbstractRange, Vararg{Any}})
!isempty(intersect(A[1], B[1])) ? _indicesmightoverlap(tail(A), tail(B)) : false
end
# But in the common AbstractUnitRange case, there's an even faster shortcut
@inline function _indicesmightoverlap(A::Tuple{AbstractUnitRange, Vararg{Any}}, B::Tuple{AbstractUnitRange, Vararg{Any}})
max(first(A[1]),first(B[1])) <= min(last(A[1]),last(B[1])) ? _indicesmightoverlap(tail(A), tail(B)) : false